Improved study of the $\beta$-function of $SU(3)$ gauge theory with $N_f=10$ massless domain-wall fermions

Abstract

I perform an improved study of the $\beta$-function of $SU(3)$ lattice gauge theory with $N_f=10$ massless optimal domain-wall fermions in the fundamental representation, which serves as a check to what extent the scenario in the previous work [arXiv:1603.08854; Proc. Sci. LATTICE2016 (2017) 228] is valid. In the finite-volume gradient flow scheme with $c=\sqrt{8t}/L=0.3$, the renormalized couplings $g^2(L,a)$ of four primary lattices ($L/a=\{8,10,12,16\}$) are tuned (in $6/g_0^2$) to the same $g_c^2$ with statistical error less than $0.5\%$, in contrast to the previous work where $g^2(L,a)$ were obtained by the cubic-spline interpolation. Then the renormalized couplings $g^2(sL,a)$ of the scaled lattices ($sL/a=\{16,20,24,32\}$ with $s=2$) are computed at the same $6/g_0^2$ of the corresponding primary lattices. Using the renormalized couplings of four lattice pairs $(sL,L)/a=\{(16,8),(20,10),(24,12),(32,16)\}$, the step-scaling $\beta$-function $[g^2(sL,a)-g^2(L,a)]/\ln (s^2)$ is computed and extrapolated to the continuum limit $\beta (s,g_c^2)$, as summarized in Table III. Based on the four data points of $\beta (s,g_c^2)$ at $g_c^2=\{6.86(2),6.92(3),7.03(2),7.16(2)\}$, I infer that the theory is infrared near-conformal, or conformal with the fixed-point $g_{\ast }^2=7.55(36)$. This corrects the scenario in the previous work with $g_{\ast }^2\sim 7.0$, and also suggests that the interpolation method cannot give a reliable determination of the $\beta$-function, especially in the regime close to the infrared fixed-point.

Comments: 22 pages, 3 tables, 4 figures, v3: published version